On Load Balancing in a Dense Wireless Multihop Network

Transcription

1 1 On Load Balancing in a Dense Wieless Multihop Netwok Esa Hyytiä and Joma Vitamo Netwoking Laboatoy, Helsinki Univesity of Technology, P.O. Box 3, FIN 215 TKK, Finland bstact We study the load balancing poblem in a dense multihop netwok, whee a typical path consists of lage numbe of hops, i.e., the spatial scales of a typical distance between souce and destination, and mean distance between the neighbouing nodes ae stongly sepatated. In this limit, we pesent a geneal famewok foanalysingthetafficloadesultingfomagivenset of paths and taffic demands. We fomulate the load balancing poblem as a minmax poblem and give two lowe bounds fo the achievable minimal maximum taffic load. The famewok is illustated by an example of unifomly distibuted taffic demands in a unit disk with a few families of paths given in advance. With these paths we ae able to decease the maximum taffic load by facto of 33 4% depending on the assumptions. The obtained taffic load level comes quite nea the tightest lowe bound. Index Tems multihop netwok, load balancing I. INTRODUCTION In a dense wieless multihop netwok a typical path consists of seveal hops and intemediate nodes along a path act as elays. In this pape we focus on studying the taffic load in such a netwok. By taffic load we mean, oughly speaking, the ate at which packets ae tansmitted in the poximity of a given node, and the objective of load balancing is to find such paths that minimise the maximum taffic load in the netwok. In paticula, we assume a stong sepaation in spatial scales between the macoscopic level, coesponding to a distance between the souce and destination nodes, and the micoscopic level, coesponding to a typical distance between the neighbouing nodes. This assumption justifies modelling the outes on the macoscopic scale as smooth geometic cuves as if the undelying netwok fabic fomed a homogeneous, continuous medium. The micoscopic scale coesponds to a single node and its immediate neighbous. t this scale the above assumptions imply that only the diection in which a paticula packet is tavesing is significant. In paticula, consideing one diection at a time thee exists a cetain maximum flow of packets a given MC potocol can suppot (packets pe unit time pe unit length, density of pogess ). Geneally, this maximal sustainable diected packet flow depends on the paticula MC potocol defining the scheduling ules and possible coodination between the nodes. Detemining the value of this maximum is not a topic of this pape but is assumed to be given (known chaactestic constant of the medium). By a simple time shaing mechanism this maximal value can be shaed between flows popagating in diffeent diections. s a esult, the scala o total flux (to be defined in Section II) of packets is bounded by the given maximum, and the load balancing task is to detemine the paths in such a way that the maximum flux in minimised. Unde the assumption of a dense multihop netwok the shotest paths (SP) ae appoximately staight line segments [PP3]. Staight paths yield an optimal solution in tems of mean delay when the taffic demands ae low and thee ae no queueing delays. Howeve, they typically concentate significantly moe taffic in the cente of netwok than elsewhee, and as the taffic load inceases the packets going though the cente of the netwok stat to expeience queueing delays and eventually the system becomes unstable when the maximal sustainable scala flux is exceeded. Hence, the use of shotest paths limits the capacity of the multihop netwok unnecessaily and ou task is to minimise the maximum packet flux in the netwok by a pope choice of paths on the macoscopic scale. Ou main contibution is the fomulation of the taffic load and the coesponding load balancing poblem in a dense multihop netwok. Fo the load balancing poblem we povide two lowe bounds. Futhe, we show how the scala flux can be calculated fo a given set of cuvilinea paths. Even though the esults ae valid only in the limit of a dense netwok (i.e., a lage numbe of nodes and a small tansmission ange), they give insight and can seve as useful appoximations fo moe ealistic scenaios. The est of the pape is oganised as follows. In Section II we pesent the necessay mathematical famewok. In Section III two lowe bounds fo the achievable

2 2 taffic load level ae pesented. In Section IV the geneal expessions fo the taffic load with cuvilinea paths ae deived. In Section V we demonstate the load balancing in unit disk with thee diffeent path sets yielding a bette pefomance than the shotest paths in tems of maximum taffic load. Section VI contains ou conclusions.. Related Wok In [PP3] Pham et al., and late in [GK4] Ganjali et al., have studied the load balancing using multipath outes instead of shotest paths. The analysis is done assuming a disk aea and a high node density so that the shotest paths coespond to staight line segments. In multipath situation the staight line segments ae eplaced by ectangula aeas whee the width of the ectangle is elated to the numbe of multiple paths between a given pai of nodes. In paticula, multiple paths ae fixed on both sides of the shotest path. In [DBT5] Dousse et al. study the impact of intefeence on the connectivity of lage ad hoc netwoks. They assume an infinite aea and the behaviou of each node to be independent of the othe nodes, which, togethe with intefeence assumptions, define the stochastic popeties fo the existance of links. With these assumptions the authos study the existance of a gigantic component, which is elated to the netwok connectivity. In [SMS5] Sikeci-Megen et al. study a dense wieless netwok with coopeative elaying, whee seveal nodes tansmit the same packet simultaneously in ode to achieve a bette signal-to-noise atio. In the analysis an infinitely long stip is studied and the authos ae able to identify a so-called citical decoding teshold fo the decode above which the message is pactically tansmitted to any distance (along the stip). The analysis assumes a dense netwok similaly as in this pape. In a dense netwok with shotest path outing the tansmission of each packet coesponds to a line segment in the aea of the netwok. line segment pocess with unifomly distibuted end points is simila to the so-called andom waypoint (RWP) mobility model commonly used in studies of wieless ad hoc netwoks [JM96], [BW2], [BRS3], [NC4]. In the RWP model the nodes move along staight line segments fom one waypoint to the next and the waypoints ae assumed to be unifomly distibuted in some convex domain. The similaity between the RWP pocess and the packet tansfes with the shotest path outes is stiking and we can utilise the eadily available esults fom [HLV5] in this case. Fo cuvilinea paths the situation, howeve, is moe complicated and the new esults deived in the pesent pape allow us to compute the esulting scala packet flux (i.e., taffic load). II. PRELIMINRIES In this section we intoduce the necessay notation and definitions fo analysing the tanspot of the packets and the esulting taffic load in the netwok. Let denote the egion whee the netwok is located. The packet geneation ate coesponding to taffic demands is defined as follows. Definition 1 (taffic demands) The ate of flow of packets fom a diffeential aea element d about 1 to a diffeential aea element d about 2 is λ( 1, 2 ) d 2 ( taffic matix ). Remak 2 The total packet geneation ate is given by Λ= d 2 1 d 2 2 λ( 1, 2 ). Each geneated packet is tansfeed along a cetain multihop path. Moe fomally, Definition 3 Set of paths denoted by P defines fo all souce destination pais ( 1, 2 ) a unique path p P. Futhemoe, let s(p, 1, 2 ) denote the distance fom 1 to 2 along the path p. Remak 4 The mean path length, i.e., the mean distance a packet tavels, is given by l = 1 d 2 1 d 2 2 λ( 1, 2 ) s(p, 1, 2 ) Λ Example 5 Fo the shotest paths we have l sp = 1 d 2 1 d 2 2 λ( 1, 2 ) 2 1. Λ Pobably the most impotant quantity fo ou puposes is the packet aival ate into the poximity of a given node. This is descibed by notion of scala flux, which in tun is defined in tems of angula flux. These ae simila to coesponding concepts of paticle fluxes in physics, e.g., in neuton tanspot theoy [BG7]. In ou case, the packet fluxes depend on the taffic demands λ( 1, 2 ) and the chosen paths P, and ae defined as follows: Definition 6 ngula flux of packets at point in diection θ, denoted by Φ(, θ) = Φ(P,, θ), is equal to the ate at which packets flow in the angle inteval (θ, θ + dθ) acoss a small line segment of the length ds pependicula to diection θ at point divided by dx dθ in the limit dx and dθ.

3 3 Φ(,θ) x x a 2 a 1 θ Fig. 1. ngula flux fom diection θ multiplied by x gives a specific packet aival ate to a box with side length x. Fig. 2. Notation fo node pdf (2) of the RWP model. Definition 7 Scala flux of packets at point is Φ() =Φ(P, ) = 2π Φ(P,,θ) dθ. Remak 8 Scala flux of packets is equal to the ate at which packets ente a disk with diamete d at point divided by d in the limit when d. The poof follows tivially fom the definitions. lso, in analogy with paticle flux in physics the following holds. Remak 9 Fo scala flux Φ(), packet density n() and constant (local) velocity v() it holds that Φ() =n() v(). (1) Poof: Packet aival ate fom a diection inteval (θ, θ + dθ) acoss a line segment pependicula to θ at with length x is equal to Φ(,θ) x dθ (cf., Fig. 1 and Def. 6). Each packet spends time x/v() inside the box, and, accoding to Little s esult, the mean numbe of packets in the box aiving fom diection (θ, θ + dθ) is Φ(,θ) x 2 /v() dθ contibuting an amount of Φ(,θ)/v() dθ to the packet density. Integating ove θ gives n() =Φ()/v(). Example 1 With unifom taffic demands and shotest path outing the esulting system is closely elated to the andom waypoint model (RWP). In the basic RWP model a node moves fom one waypoint to anothe along the staight line segment and the waypoints ae unifomly distibuted in some convex domain with aea. The stationay node distibution is given by [HLV5] f RWP () = 1 2l 2 2π dθ a 1 a 2 (a 1 + a 2 ), (2) whee l is the mean leg length, a 2 = a 2 (,θ) is the distance to the bounday in the diection θ and a 1 in the opposite diection (see Fig. 2). With v =1the leg geneation ate is equal to 1/l, and hence the stationay node pdf of the RWP model is identical to the packet density n() with an appopiate scaling, n() =l f RWP () Λ. With v = 1 we have Φ() = n() and the flux with unifom taffic demands and shotest paths is given by, Φ() = Λ 2 2 2π dθ a 1 a 2 (a 1 + a 2 ). With this notation we can finally give a fomal definition fo the optimisation poblem. Definition 11 (load balancing poblem) Find the set of paths, P opt, minimising the maximum scala flux, P opt =agmin P max Φ(P, ). Remak 12 (optimal maximum taffic load) With the load balanced paths the maximum load is Φ opt =max Φ(P opt, ) =min P max Φ(P, ). In the above defintion, Def. 11, one needs the scala flux Φ(P, ). We will show in Section IV how this can be calculated fo a given set of paths P. Finding the optimal paths is a difficult poblem of calculus of vaiation. In this pape, we do not seach fo a geneal solution but athe study thee heuistically chosen families of paths and compae thei pefomance with that of the shotest paths and with the bounds intoduced in the next section. III. LOWER BOUNDS FOR PCKET FLUX Ou next goal is to deive two lowe bounds fo achievable load balancing, i.e., fo a given taffic patten λ( 1, 2 ) we want to find bounds fo the minimum of the maximal taffic load that can be obtained by a pope choice of paths. Let us stat by an illustative example. Example 13 Conside an h w ectangle whee packets ae geneated at ate Λ at the bottom of the aea (unifomly) as illustated in Fig. 3. ll packets tavel diectly up to the top of the aea, i.e., we have shotest paths with mean path length of l = h. The aival ate of packets acoss any hoizontal line segment of length t is equal to Λ t/w. Hence, the flux at any point is constant, Φ=Λ/w =Λl/, and we have a pefect load balancing (cf. Def. 11).

4 4 destinations souces h = l w Fig. 3. Packets moving unifomly fom the bottom to top of the aea yield a unifom scala flux of Φ=Λ/w =Λ l/. ds dθ θ dθ x θ x θ x x s h x d θ Note that the quantity Λ l t coesponds to the cumulative distance the packets aiving duing a time inteval of t have to tavel on aveage in ode to each thei espective destinations. Consequently, the pevious example suggests the following poposition. Poposition 14 (distance bound) max Φ(P, ) Λ l. (3) Poof: Without loss of geneality we can set v =1 whence Φ() =n(). Let be the aea of and n the mean density of packets, n = N/,wheeN is the mean numbe of packets in the system. With v =1, Little s esult implies N =Λ l. Hence max Φ() n = N = Λ l. Remak 15 Fo P sp consisting of staight line segments between the souce destination pais, we obtain the lowest possible value fo l = l sp. Consequently, Φ opt Λ l sp. (4) ltenatively, we can only conside taffic flows cossing an abitay bounday (cf., cut bound in wied netwoks). Poposition 16 (cut bound) Fo any cuve C which sepaates the domain into two disjoint domains 1 and 2 it holds that Φ opt 1 L 1 d d 2 2 (λ( 1, 2 )+λ( 2, 1 )), whee L is the length of the cuve C. Poof: Conside fist a shot line segment ds at at some point along the cuve C. Letγ denote a diection pependicula to the cuve at such that the packets aiving fom the angles (γ π/2,γ + π/2) coss ds fom outside to inside, and packets aiving fom (γ + Fig. 4. Deivation of expession fo the scala flux. π/2,γ+3π/2) coss ds fom inside to outside. The ate at which packets move acoss ds is clealy given by π/2 λ() ds = cos α (Φ(,γ+α)+Φ(,γ+α+π)) dα ds. π/2 s cos α 1 fo π/2 α π/2 we get λ(ds) π/2 π/2 Φ(,γ+ α)+φ(,γ+ α + π) dα ds =Φ() ds max Φ(x) ds. x Integating ove the cuve C completes the poof. IV. PCKET FLUX WITH CURVILINER PTHS In this section, unless stated othewise, we assume unifom taffic demands and a single path p( 1, 2 ) between souce and destination locations 1 and 2. Moeove, we assume the paths in P satisfy the so-called path continuity constaint: Definition 17 (path continuity) If p( 1, 2 ),thenp( 1, 2 )=p( 1, )+p(, 2 ). The above definition lets us chaacteise the paths accoding to the diection at some point x. In paticula, the outing decision made in each point depends only on the destination of the packet, not the souce. Let p(x,θ) denote a path going though point x and having a diection θ at that point. The points along the cuve (assumed to be smooth) ae denoted by p(x,θ,s), whee s [ a 1,a 2 ],anda 1,a 2 >, so that p(x,θ,) = x. Thus, a 1 and a 2 denote the distance to the bounday along the path in opposite diections. Note that this means that we limit ouselves to paths that stat and end at the bounday of the domain (no closed paths within the domain allowed).

5 5 Definition 18 (path divegence) Let h(x,θ,s) denote the ate with espect to the angle θ at which paths divege at the distance of s, p(x,θ+ dθ, s) p(x,θ,s) h(x,θ,s) = lim dθ dθ = θ p(x,θ,s). Poposition 19 (angula flux with cuvilinea paths) Fo a unifom taffic demands, λ( 1, 2 )=Λ/ 2,the angula flux at point x in diection θ is given by Φ(x,θ)= Λ 2 a 1 h(x,θ, s ) h(x,θ,s ) h(x,θ,s+s ) ds ds. (5) Poof: Without loss of geneality we may assume Λ = 1. Ou aim is to detemine the angula flux in diection θ denoted by Φ(x,θ). To this end assume that a paticula souce is located in a diffeential aea element about point x (see Fig. 4 left) x = p(x,θ,s ), s, fo which it clealy holds that p(x,θ,s s )=p(x,θ,s). Let dθ denote a diffeential angle at point x so that the diffeential souce aea about x is given by (see Fig. 4 left) s = h(x,θ,s ) dθ ds. Similaly, let dθ denote a small angle at point x,which yields a destination aea of d = h(x,θ,s s ) ds dθ, as illustated in Fig. 4 (ight). The height of the taget line segment pependicula to the path at point x is h x = h(x,θ, s ) dθ. Hence, the contibution to the angula flux fom the diffeential souce aea is dφ = s d 2 dθ h x = 1 ( ) 2 1 dθ 1 h(x,θ, s ) dθ ) (h(x,θ,s a 2 ) dθ ds h(x,θ,s s ) ds dθ = 1 2 h(x,θ,s ) h(x,θ, s ) h(x,θ,s s ) ds ds. Hence, the angula flux at x in diection θ is given by Φ(x,θ)= 1 2 a 1 h(x,θ,s ) h(x 1,θ, s ) h(x,θ,s s ) ds ds. The poposition follows upon substition s s. Remak 2 (angula flux with non-unifom λ( 1, 2 )) It is staightfowad to genealise (5) to the case of non-unifom taffic demands λ( 1, 2 ). In this case, the angula flux at point x in diection θ is given by Φ(x,θ)= a 1 h(x,θ, s ) h(x,θ,s ) λ(x, p(x,θ,s+s )) h(x,θ,s+s ) ds ds. Example 21 Fo the shotest paths we have h(x,θ,s)= s, and consequently, fo unifom taffic demands, the angula flux is given by Φ(x,θ)= Λ 2 = Λ 2 a 1 a 1 (s + s ) ds ds a 2 2 /2+a 2s ds = Λ 2 2 a 1a 2 (a 1 + a 2 ), in accodance with [HLV5] as pesented in Ex. 1. V. EXMPLE: UNIT DISK WITH UNIFORM DEMNDS In this section we will demonstate how the poposed famewok can be applied. To this end, we conside a special case of a unit disk with unifom load, = { R 2 : < 1}, and, λ( 1, 2 )=Λ/π 2. We study the pefomance of thee simple families of paths: oute and inne adial ing paths and cicula paths. The pefomance of these path sets is compaed with that of the shotest paths, and with the appopiate lowe bounds fo the minimal maximum taffic load. Example 22 (unit disk with shotest paths) Fo tanspot accoding to the staight line segments we can ely on the esults fo the RWP model (see Ex. 1 and [HV5]). t distance d, the scala flux is given by Φ sp (d) = 2(1 d2 ) Λ π 2 π 1 d 2 cos 2 φdφ.

6 6 In paticula, the maximum flux is obtained at the cente, Φ sp () = 2 Λ.6366 Λ. (6) π Example 23 (distance bound fo unit disk) The distance bound gives us a elationship between the obtainable maximum load and the mean path length. With shotest paths we have l sp = 128/45π which upon substitution in (4) yields Φ opt Λ π Λ. Example 24 (uppe bound fo mean path length) The known taffic load when shotest paths ae used can be combined with the distance bound. ccoding to (6) Φ sp = 2Λ π = Λ 2. On the othe hand, accoding to (3) we have max Φ(P, ) Λ l =Φ sp l 2. Hence, if with the given paths P the mean path length l>2, then the esulting maximum taffic load is geate than the one obtained with the shotest paths. Example 25 (cut bound) Let the sepaating cuve C be a concentic cicle with adius d, <d<1. Fothe packet ate acoss the bounday it holds that λ(d) 2d 2 (1 d 2 ) Λ, which coesponds to adial flux (pe unit length) Φ (d) = 2d2 (1 d 2 ) 2πd Λ= d d3 π Λ Φ opt. s this is a lowe bound fo the scala flux we want to maximise it. The tightest lowe bound is obtained with d =1/ 3, 2 Φ opt 3 Λ.1225 Λ. 3 π Hence, in this case, if compaing to the shotest paths, the cut bound gives a lowe bound which says that no moe than 8% impovement in the maximum load is possible, while accoding to the distance bound at most a 55% impovement is possible.. Radial Ring Paths Let us next conside thee actual path sets as illustated in Fig. 5. The shotest paths (SP) ae equivalent to RWP model as has been aleady mentioned. The two adial path sets ae simila in that each path consists of two sections. One section is a adial path towads (o away fom) the oigin, and the othe section is an angula path along a ing with a given adius. The diffeence between the two sets is in the ode of components, Rout uses the oute angula ings and Rin the inne ones, as the names suggest. Note that locally, at any point, the packets ae tansmitted only in 4 possible diections (2 adial and 2 angula), which may simplify the possible implementation of the time division multiplexing. When studying the aival ate into a small aea at the distance of d fom the oigin one needs to conside both adial and angula ing movement. The adial component of the flux is the same fo both path sets, i.e., Φ (d) = d d3 Λ. (7) π 1) Oute adial ing paths: Let us next conside oute adial ing paths. We want to detemine the flux along the ing at the distance of d. To this end, conside a small line segment fom ( d, ) to ( d, ) as taget aea as illustated in Fig. 6. Packets oiginating fom a small souce aea at the distance of d in diection θ tavel though the taget line segment if thei destination is in destination aea. The size of the souce aea is d dθ, while the possible destination aea is θ πd 2 2π = d2 2 θ. Combining these with (7) yields the angula flux at the distance of d, Φ θ (d) = 4Λ π d 2 π 2 2 θ dθ = d3 Λ. Hence, the total flux at the distance of d fo oute path set is given by Φ Rout (d) =Φ (d)+φ θ (d) = (π 1)d3 + d π The maximum flux is obtained at point d =1, Φ Rout (1) = Λ. Λ.

7 7 souce Rout 1 Rin SP.8.6 SP Rout destination.4 SP Rout.2 Rin Fig. 5. Left figue illustates the thee path sets consideed: staight line segments (SP), adial paths with oute (Rout) and inne (Rin) angula ing tansitions. In ight figue the esulting flux is plotted fo fo the thee path sets (SP, Rout and Rin) and fo a andomised combination of SP and Rout (dashed cuve) as a function of distance d fom the cente. taget d souce dθ θ Fig. 8. points. Cicula paths illustated fo diffeent destination (o souce) destinations Fig. 6. Deivation of the angula ing flux at the distance of d fo oute adial ing paths. 2) Inne adial ing paths: Fo inne paths the possible destination aea of packets is and Φ θ (d) = 4Λ π π 2 = 2(d d3 ) Λ π 2 1 d 2 θ, 2 1 d 2 θ d dθ 2 π θdθ=(d d 3 ) Λ. Consequently, combining the above with (7) gives Φ Rin (d) =(1+1/π)(d d 3 ) Λ. The maximum is obtained at point d =1/ 3, Φ Rin (1/ 3).57 Λ. Hence, the oute vesion leads to a highe maximum load than the shotest paths while the inne vesion yields a slightly bette solution. The esulting packet fluxes ae illustated in Fig. 5 (ight) as a function of the distance d fom the cente. B. Cicula paths s the last path set we conside cuvilinea paths, efeed to as cicula paths, which consist of such sections of cicumfeence of cicles that coss the unit disk at the opposite points as illustated in Fig. 7. Some example paths ae illustated in Fig. 8 fo diffeent destination (o souce) locations. Fom the figue it can be seen that these paths smoothly move some potion of the taffic away fom the cente of the aea. The angula flux can be calculated using Poposition 19, and the scala flux is obtained by integation (cd. Definition 7). The esulting scala flux is depicted in Fig. 7 (ight). It can be seen that the taffic load is faily well distibuted in the aea. The maximum flux is obtained at the cente of the disk, whee the flux is In fact, it is possible to detemine the packet flux at the cente analytically. Fo this we have Φ() = 4 Λ, (.4244 Λ) 3π which is exactly 2/3 of the flux with the shotest paths consisting of staight line segments (cf. Ex. 22). C. Randomised path selection appoach One option to achieve a lowe maximum load is to allow the use of seveal paths fo each pai of nodes (similaly as in [PP3], [GK4]). In paticula, let us elax ou assumptions a bit and assume a finite numbe of path sets {P i },wheei =1,...,n. Upon tansmission of a packet the souce node chooses a path fom path set i with pobability of i, i =1,...,n.

8 Fig. 7. Cicula paths ae paths fomed by the cicumfeences of cicles which coss the unit disk at the opposite points. The figue on ight illustates the esulting flux as a function of distance fom the cente. Poposition 26 (packet flux with andomised paths) andomised path selection fom path sets {P i }, i =1,...,n, upon tansmission with pobabilities p i, i =1,...,n yields a total scala flux of Φ() = i p i Φ(P i, ). Example 27 Conside unifom taffic demands in unit disk and two path sets, 1) shotest paths, and 2) the oute adial paths. Weights p 1 =.6 and p 2 =.4 give a packet flux of Φ(d) =.61 Φ sp (d)+.39 Φ Rout (d). The esulting flux is almost constant as illustated by the dashed line in Fig. 5. The maximum is.397 Λ, which is about 37% lowe than with the shotest paths. Example 28 The same technique can be taken futhe, e.g., by combining all thee path sets as follows Φ(d) =.52 Φ sp (d)+.37 Φ Rout (d)+.11 Φ Rin (d), which gives a maximum flux of.379 Λ, i.e., about 4% loweing in the maximum flux when compaed to the shotest paths. VI. CONCLUSIONS In this pape we have pesented a geneal famewok fo analysing taffic load and outing in a lage dense multihop netwok. The appoach elies on stong sepaation of spatial scales between the micoscopic level, coesponding to the node and its immediate neighbous, and the macoscopic level, coesponding to the path fom the souce to the destination. In a dense wieless netwok with this popety the local taffic load can be assimilated with the so-called scala packet flux. The packet flux is bounded by a maximal value that the netwok with a given MC and packet fowading potocol can sustain. The packet flux depends on taffic patten λ( 1, 2 ) and the chosen set of outing paths P. Theload balancing poblem thus compises of detemining the set of outing paths such that the maximal value of the flux in the netwok is minimised. While the geneal solution of this difficult poblem emains fo futue wok, ou main contibution in this pape consists of giving bounds fo the packet flux and giving a geneal expession fo detemining the packet flux at a given point when a given set of cuvilinea paths is used. The esults ae illustated by numeical examples with thee diffeent sets of paths in unit disk. Futue wok includes investigating how to find nealy optimal load balancing in a distibuted fashion. REFERENCES [BG7] G.I. Bell and S. Glasstone. Nuclea Reacto Theoy. Reinhold, 197. [BRS3] C. Bettstette, G. Resta, and P. Santi. The node distibution of the andom waypoint mobility model fo wieless ad hoc netwoks. IEEE Tansactions on Mobile Computing, 2(3): , July Septembe 23. [BW2] C. Bettstette and C. Wagne. The spatial node distibution of the andom waypoint mobility model. In Poc. of Geman Wokshop on Mobile d Hoc netwoks (WMN), Ulm, Gemany, Mach 22. [DBT5] Olivie Dousse, Fançois Baccelli, and Patick Thian. Impact of intefeences on connectivity in ad hoc netwoks. IEEE/CM Tans. Netwoking, 13(2): , pil 25. [GK4] Yasha Ganjali and btin Keshavazian. Load balancing in ad hoc netwoks: single-path outing vs. multi-path outing. In Poc. of Infocom 4, Hong Kong, Mach 24. [HLV5] Esa Hyytiä, Pasi Lassila, and Joma Vitamo. Spatial node distibution of the andom waypoint mobility model with applications. IEEE Tansactions on Mobile Computing. to appea in. [HV5] Esa Hyytiä and Joma Vitamo. Random waypoint mobility model in cellula netwoks. Spinge Wieless Netwoks, 25. to appea in. [JM96] David B. Johnson and David. Maltz. Dynamic souce outing in ad hoc wieless netwoks. In Tomasz Imielinski and Hank Koth, editos, Mobile Computing, volume 353, [NC4] [PP3] pages Kluwe cademic Publishes, W. Navidi and T. Camp. Stationay distibutions fo the andom waypoint mobility model. IEEE Tansactions on Mobile Computing, 3(1):99 18, Januay-Mach 24. Pete P. Pham and Sylvie Peeau. Pefomance analysis of eactive shotest path and multi-path outing mechanism with load balance. In Poc. of Infocom 3, volume 1, pages , San Fancisco, US, Mach-pil 23. [SMS5] Bisen Sikeci-Megen and nna Scaglione. continuum appoach to dense wieless netwoks with coopeation. In Poc. of Infocom 5, Miami, FL, US, 25.